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If a right triangle is inscribed in a circle, then the hypotenuse is a diameter. Conversely, if one side of an inscribed triangle's sides is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
These statements are commonly referred to as Thales' Theorem and the converse of Thales' Theorem.
Consider a right triangle ABC, with m∠ B = 90^(∘), inscribed in a circle.
According to the Inscribed Angle Theorem, the measure of ∠ B is half the measure of its intercepted arc, AC.
Substituting m∠ B=90^(∘) into the equation above gives that m AC=180^(∘). Then, AC is a semicircle, implying that AC (the hypotenuse of △ ABC) is a diameter of the circle.
Consider a triangle ABC inscribed in a circle such that one side of the triangle is a diameter of the circle.
Since AC is a diameter, then AC is a semicircle and then mAC=180^(∘). Now, the Inscribed Angle Theorem gives a relation between this arc and the angle opposite to the diameter. m∠ B &= 1/2mAC [0.15cm] &= 1/2(180^(∘)) &= 90^(∘) This implies that ∠ B is a right angle, which makes △ ABC a right triangle.